Answer:
y = -3(x +4)(x -1)/((x +3)(x -6))
Step-by-step explanation:
You want a rational function with vertical asymptotes at -3, 6; x-intercepts at -4, 1; and a horizontal asymptote at y = -3.
A rational function will have vertical asymptotes at the denominator zeros. Vertical asymptotes at -3 and 6 will be had if the denominator has factors (x +3) and (x -6).
A rational function will have x-intercept at the numerator zeros. X-intercepts at -4 and 1 will be had if the numerator has factors (x +4) and (x -1).
A horizontal asymptote will be found at the y-value that is the (constant) ratio of the highest-degree terms of the numerator and denominator. Here, we can achieve that by multiplying the factors mentioned above by the desired constant (-3).
The equation is ...
[tex]\boxed{y=-3\dfrac{(x+3)(x-6)}{(x+4)(x-1)}}[/tex]
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Additional comment
A "hole" will be found where a numerator and denominator factor cancel. For example, a hole at x=7 could be achieved by making the function have an additional numerator factor of (x -7) and an additional denominator factor of (x -7).
Effectively, the horizontal asymptote is the quotient of the numerator and denominator (excluding the "remainder"). If that quotient is a linear (or other) function, the graph will asymptotically approach that function as the magnitude of x gets large. A linear function gives a "slant aymptote."
